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arXiv:cond-mat/0610828v1 [cond-mat.mtrl-sci] 30 Oct 2006

Adiabatic and Nonadiabatic Electronics of Materials

V.A. Vdovenkov1

1Moscow State Institute of Radioengineering,

Electronics and Automation (technical university)

Vernadsky ave. 78, 117454 Moscow, Russia

Abstract

Physical foundations of adiabatic and nonadiabatic electronics of materials are considered in this

article. It is shown the limitation of adiabatic approach to electronics of materials. It is shown that

nonadiabatic physical properties of solid materials (hyperconductivity, superconductivity, thermal

superconductivity and phonon drag of electrons at Debye’s temperatures of phonons) depend on

oscillations of atomic nuclei in atoms.

PACS numbers: 63.20.Kr, 63.90.+t, 71.00.00, 72.20.Pa

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I. INTRODUCTION

In modern electronics of materials, adiabatic principle mainly is used. According to

this principle, exchanging of energy between electrons and nuclei of atoms are neglected.

Today’s dominating electronics in essence is adiabatic electronics. It allows to consider

materials approximately, in adiabatic approach, and to study limited circle of their physical

properties. Development of nonadiabatic electronics, on the contrary, just begins. It takes

into the account processes of energy exchange by between electrons and nuclei of atoms, has

no basic restrictions, can reduce or remove problems of modern electronics and reveal new

physical properties of materials.

This article is devoted to analysis of physical foundations of adiabatic and nonadiabatic

electronics of materials and properties of boson-fermion fluid, arising under nonadiabatic

conditions in materials.

II. ADIABATIC ELECTRONICS OF MATERIALS

Bases of adiabatic electronics were stated by M. Born and R. Oppenheimer [1] in the

solution of Shrodinger’s stationary equation for a material HΨ = WΨ, where Hamiltonian

H contains operators of kinetic energies of electrons Te and atomic nuclei Tz , and also

the crystal potential V is dependent on sets of electrons coordinates (r) and atomic nuclei

coordinates (R) in a material. According to M. Born and R. Oppenheimer this equation is

equivalent to the following two equations:

(Te+ V )φ(r,R) = Eφ(r,R), (1)

(Tz+ E + A)Φ(R), (2)

where Te- kinetic energy of electrons’ system, W - energy of a material, φ(r,R) and Φ(R)

- wave functions for electrons system and nuclei system,

A = −

?

I

(¯ h2/MI)

?

φ∗(r,R)∇2

Rφ(r,R)dτ (3)

- adiabatic potential, MI- mass of nucleus in I −th atom, dτ - an element of the material’s

volume. Potential A describe exchange of energy between electrons and nuclei. In adiabatic

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approach, potential A is small comparing to W, and it is neglected. One believe that if A ≡ 0

then the exchange of energy between electrons and nuclei is impossible, then, one believes

that the problem of a material, stated by the equations (1, 2), is an adiabatic problem, and

electronic properties, described by the equation (1), represent a theoretical basis of adiabatic

electronics of materials. However, one can see from equations (1, 2) that upon exception of

A, a full separation of variables in these equations is not achieved, as r and R are arguments

of V . Hence, the exchange of energy between systems of electrons and nuclei in adiabatic

approach, generally speaking, is not excluded.

So, physical parameters of crystalline silicon were calculated in adiabatic approach by G.

Pastore, E. Smargiassi and F. Buda [2]. It turned out, that kinetic and potential energies

of electrons system and nuclei system deviate from average values in opposite phases to

each other with characteristic frequencies.It means that the energy exchange between

electrons and nuclei system occurs even at adiabatic conditions. Autors [2] supposed that the

greatest among the frequencies was the result of numerical method of calculation, because

at that time such frequency oscillations in crystals were not known. Later it was found

out, that this frequency coincides with the sum of two frequencies (frequency of inherent

oscillations of nucleus in silicon atom minus frequency of crystal phonon). In other words,

inherent oscillations of atomic nuclei are the fundamental property of manerials. Amplitudes

of nucleus oscillations are close to 10−12meter and consequently processes connected to

them refer to subangstroem electronics of materials. These oscillations in particular can

be an energy source for chemical reactions and for phase transitions even at extremely low

temperatures. The given result allows us to consider the adiabatic principle differently.

Adiabatic condition (A ≡ 0) means impossibility of the systematical, unidirectional stream

of energy from electrons to nuclei or from nuclei to electrons only, but does not exclude the

stream of energy periodically changing its direction.

P. Dirac [3] investigated the time-dependent Schrodinger’s equation for a crystal and

studied an opportunity of co-ordinates separation for electrons and nuclei that correspond

to adiabatic principle. He has shown that electrons and atomic nuclei move within the

effective self-consisted potential fields and generally their movements cannot be described

by equations, independent from each other. Therefore, the adiabatic principle in materials

is approximate, permitting to use it with some accuracy in certain conditions, and its ap-

plying each time requires substantiation and estimation of possible mistakes arising during

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calculation of physical quantity in such an approach.

The regular error of energy calculation for electrons in adiabatic approximation is about

< A >≈ 10−5W. It turns out to be comparable with widths of forbidden energy gapes in

many semiconductors and therefore is not always allowable. In particular, the given error

can be one of the reasons of inexact definition of deep energy levels of the local centers in

semiconductors, calculated under adiabatic approximation.

The amendments to energy of the crystal, that arise as a result of removal of potential

eq. (3) out of equation (2), according to M. Born [1], are proportional to the integer powers

of small parameter η = (m/MI)1/4<< 1 , where m- electron mass. It has given an occasion

to justify any applying of adiabatic approximation, based on smallness of η, proved to be

wrong, and, as a matter of fact, it is generally wrong in all cases. Nonadiabatic principle

and terms of its applying to materials have other physical sense.

III. CONDITIONS FOR APPLYING ADIABATIC APPROACH

It was shown by C. Herring [4] (see else: [5]), that adiabatic approach can be applied, if

Ekl>>

?

µ

¯ hωµ∆Rµ

?

d3r φ∗

k

∂

∂Rµ

φl ,(4)

where Ekl - energy of the allowed electronic transitions between states k and l , ∆Rµ -

characteristic displacement of nuclei system on frequency ωµ, µ - type of oscillations, φk

and φl- electron wave functions, ¯ h - Dirac’s constant. Adiabatic approach was studied by

A. Davidov [6] in conditions when oscillations of atomic nuclei are allowed only with one

frequency ωµ. He has come to a conclusion that adiabatic approach can be used if

Ekl>> ¯ hωµ. (5)

In other words, the adiabatic principle can be soundly applied if energy of nuclei oscillations

is less than energy of the allowed electronic transitions, for example if the energy of nuclei

oscillations is less than width of the forbidden energy gape of a semiconductor. The condition

eq. (5) is sufficient, but is not a requirement. Therefore in some cases applying of adiabatic

approximation is justified though condition eq. (5) is not carried out.

Thus, criterion of applicability of adiabatic approach in materials is not so simple, as it

quite often looks. The adiabatic principle dominates in modern physics of materials, but

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in some cases it is used unreasonably. Implications of adiabatic theories quite often appear

unproductive because of inaccurate applying of adiabatic approach. On the contrary, using

of nonadiabatic principle allows to reveal and to use new properties of materials. So, con-

ditions for applicability of adiabatic approach stated by C. Herring [4] and A. Davidov [6]

contain various types and frequencies of nuclei oscillations. Meanwhile, these nuclei oscilla-

tions (unlike oscillations of atoms or ions) are poorly investigated. It hinders well-founded

applications of adiabatic approach and halts development of nonadiabatic electronics.

IV. TYPES AND FREQUENCIES OF INHERENT OSCILLATIONS IN ATOMS

OF MATERIALS

For determining types and frequencies of nuclei inherent oscillations in atoms, it is ex-

pedient to use model of a crystal in which each atom is represented by an electron shell

and a nucleus connected with each other by quasi-elastic force. Crossection of such a three-

dimensional model by a plane containing centers of some electronic shells is shown on Fig. 1.

Electron shells with masses m” on Fig. 1 are represented as circles, nuclei with masses M′

FIG. 1: Crossection of a three-dimensional crystal model by a plane.

are represented by dark circles in centers of electronic shells. According to analytical me-

chanics the frequency of normal oscillations of a nucleus in j − th atom can be defined if to

suppose absolutely rigid all quasi-elastic connections in considered model except one quasi-

elastic connection acting between a nucleus and shell in a j − th atom. In such conditions

the oscillations of a crystal are represented by oscillations of a nucleus relatively motionless

electronic shell of j − th atom. Physically equivalent situation is established out if mass of

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